Box 1 Rainbows
For the understanding of the refractive scattering and the rainbow phenomenon we
have to discuss the deflection function, which describes the variation of the deflection
angle, with the impact parameter b (the distance of the incident particle from
the central axis, as illustrated in fig. B1.1). The well known optical rainbow, which
occurs after refraction and one reflection of the light beam in the water droplet, has
been correctly interpreted by Descartes, who was the first to realise the importance of
the caustic, the accumulation of lightrays observed at the maximum deflection angle.
The deflection function for light in a waterdroplet is shown in fig. B1.1 together with
a case of nuclear scattering, (16O + 16O), discussed in the main text. The rainbow
phenomenon places an increased light intensity at the rainbow angle, , because a
finite region of impact parameters contributes to differentially small regions of the
deflection angle at . The position of the rainbow angle, , itself depends on the
wavelength, , because it is related to refraction, giving rise to a splitting of colours
for a white spectrum of incident waves. The maximum deflection angle would define
a sharp boundary between light and shadow - in classical optics, the intensity would
diverge at this angle as shown in fig. B1.2. The solution of the mathematical problem
is difficult and has been solved in the middle of the 20th century [1]. The results of
some mathematical approaches are illustrated in fig. B1.2. The correct mathematical
description of the intensity variation was first introduced by Airy, defining the Airy
function. The first intensity-(Airy) maximum appears slightly shifted just inside the
classical illuminated region followed by the first Airy minimum (A1) and by further
Airy maxima and minima. We will refer to these as the first, second, third etc. Airy
maxima or minima. They are observed in the data on 16O + 16O scattering at the lower
energies of 80-150 MeV.
Fig. B1.1 Deflection functions for the case of, a) the light scattering on water droplets.
In this case the deflection angle varies from total reflection at , down to
, with one reflection inside. A secondary rainbow appears (see fig 1) with a
second reflection in the water droplet. In case, b) of 16O + 16O elastic scattering at an
energy of 30 MeV/nucleon attractive potentials give rise to negative deflection angles
with a maximum at ; the rainbow angle decreases with increasing energy. For
16
O + 16O scattering the absorption at small impact parameters, (given as orbital
angular momentum L = Kb ( K = wavenumber)) is shown by the function |SL| which is
normalised to 1 at large distances. In the rainbow region the survival probability is
at a L-value of 40.
Fig. B1.2 The functional dependence of the light intensity at the maximum deflection
angle. This angle defines classically the separation between light and shadow. The
correct description is given by Airy functions, which place the intensity maximum at
angles inside the "lighted" region and exhibits oscillations, to which we will refer as
primary or secondary (and higher) Airy maxima and minima (marked as A1, A2 etc.
in figure 3). The scattered intensity now extends into the classically forbidden region
(dashed lines). The other curves denoted by Young and Descartes show unphysical
singularities at , which were later resolved by Airy.